MinT - Best Known (71−22, 71, s)-Nets in Base 8

Best Known (71−22, 71, s)-Nets in Base 8

(71−22, 71, 371)-Net over F₈ — Constructive and digital

Digital (49, 71, 371)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. digital (36, 58, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 29, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(71−22, 71, 523)-Net in Base 8 — Constructive

(49, 71, 523)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. (38, 60, 514)-net in base 8, using
  - base change [i] based on digital (23, 45, 514)-net over F₁₆, using
    - 1 times m-reduction [i] based on digital (23, 46, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 23, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
        Niederreiter sequence [i]

(71−22, 71, 1412)-Net over F₈ — Digital

Digital (49, 71, 1412)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71−22, 71, 473091)-Net in Base 8 — Upper bound on s

There is no (49, 71, 473092)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 13164 086428 973188 720477 455107 292187 430280 294234 595806 375959 591680 > 8⁷¹ [i]